Relatedly, Marcin Wichary wrote a nice post about using FFT to remove moiré and halftone effects when scanning images that were printed with halftones.
I'd like to see a sequel where the fractional fourier transform is used for image restoration
There have been some interesting advances in trying to add spectral information to the data that a learning architecture has at its disposal, but there are a couple roadblocks that I don’t think have been solved yet.
1. Complex-valued NNs are not an easy generalization of real ones.
2. A localization in one domain implies non-local behavior in the other (this is the Fourier uncertainty principle).
Fourier Neural Operators (FNOs) come close to what I want to see in this area but since they enforce sparsity in the spectral domain their application is necessarily limited.
I do wonder if a wavelet transform might be better.
I think one can do better with a wavelet, shearlet, or curvelet transform that is adapted to the problem domain at hand. But the uncertainty principle still haunts those transforms, and anyway the goal is to be domain-agile.
See also: CosAE: Learnable Fourier Series for Image Restoration (2024)
Relatedly, Marcin Wichary wrote a nice post about using FFT to remove moiré and halftone effects when scanning images that were printed with halftones.
It's from 2021: Moiré no More (https://newsletter.shifthappens.site/archive/moire-no-more/).
I'd like to see a sequel where the fractional fourier transform is used for image restoration
There have been some interesting advances in trying to add spectral information to the data that a learning architecture has at its disposal, but there are a couple roadblocks that I don’t think have been solved yet.
1. Complex-valued NNs are not an easy generalization of real ones.
2. A localization in one domain implies non-local behavior in the other (this is the Fourier uncertainty principle).
Fourier Neural Operators (FNOs) come close to what I want to see in this area but since they enforce sparsity in the spectral domain their application is necessarily limited.
I do wonder if a wavelet transform might be better.
I think one can do better with a wavelet, shearlet, or curvelet transform that is adapted to the problem domain at hand. But the uncertainty principle still haunts those transforms, and anyway the goal is to be domain-agile.
See also: CosAE: Learnable Fourier Series for Image Restoration (2024)
https://sifeiliu.net/CosAE-page/
Was there a conclusion?
[2024]